I was playing Chutes & Ladders with my four-year-old daughter yesterday, and I thought, “How long is this going to take?”
But that didn’t answer my specific question, namely, “How long is this going to take?”
So I wrote a bit of R code to simulate the game.
Here’s the distribution of the number of spins to complete the game, by number of players:
With two players, the average number of spins is 52, with a 90th percentile of 88.
If you add a third player, the average increases to 65, and the 90th percentile increases to 103. You’re playing fewer rounds, but each round is three times as long. If you add a fourth player, the average is 76 and the 90th percentile is 117.
So, in trying to minimize the agony, it seems best to not encourage my eight-year-old son to join us in the game. If he plays with us, there’s a 63% chance that it will take longer.
And that’s particularly true because then the chance of my daughter winning drops from about 1⁄2 to about 1⁄3.
That raises another question: if I let her go first, what advantage does that give her? Not much. The chance that the person who goes first will win is 50.9%, 34.4%, and 25.9%, respectively, when there are 2, 3, and 4 players. So not a noticeable amount. Thus I cheat (on her behalf). Really, though, I’m cheating in order to shorten the game as much as to ensure that she wins.